JPRS ID: 9167 TRANSLATION DYNAMICS OF ROCKET MOTOR SYSTEMS BY VE. B. VOLKOV, T.A. SYRITSYN AND G. YU. MAZIN

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JPRS L/9167
30 June 1980 _
Translation
DYNRP,~ICS OF ROCK~T MOTOR SYSTEMS
By
Ye. B. Volkov, T.A. Syritsyn and G. Yu, Mazin
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JPR5 L/9167
30 June 1980
DYNAMICS OF ROCKET MOTOR SYSTEMS
Mo~cow STATIKA I DINAMIKA RAKETNYKH DVIGATEL'NYKH USTANOVOK V
_ DWKH KNIGAKH ("5tatics and Dynamics of Rocket Motor 5ystems in F
Two Volumes") in Russian i978 signed to press 15 Mar 78 pp 1-320
[Volume Two ("Dynamics") of book by Ye. B. Volkov, T.A. Syritsyn
and G, Yu. Mazin, Mashinostroyeniye Publishers, 1550 copies;
Volume One ("Statics") was published in JPRS L/8627, 21 August
1979, entitled "Statics of Rocket Mot.or Systems"]
CONTENTS
Annotation
1
Foreword
2
Ser_tion I. Dynamic Characteristics of Liquid-Fuel ~iigines
4
Cli~pter 1. Dynamic CharacCeristics of the ~ngine Units
1.1. Thrust Chamber 6
1.1.1. PJonlinear Equa~fons 6
1.1.2. Linearized Equations of the Chamber 6
1.1.3. Dynamic-Characteristics of the Thrust Chamber 12
, 1�2. Gas Generator
1.3, Hydraulic Channels 15
1.3.1. Nonlinear Equations 13
1.3.2. Linear Equations 18
1.3.3. Consideration of the Compressibility of the 21
Liquid and the Elasticity of the Walls 23
1.4. Gas Reservoirs 25 -
1.5. Pumps 28
1.5.1. Tdonlinear Equation of the i-th Pump 28
1.5.2. Linear Equation of the Pumps 35
1.5.3. Effect of Cavitation on the Pump Characteristic 47
1.6. Turbine
1.7, Turbine Pump Assembly (TNA) 43
1.7.1. TNA [Turbine Pump Assembly] Impeller Equation 47
_ a _ [I - USSR - A FOUO]
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l.tt. Elements of the ~ngine Control System 53
1.3.1. Pneumohydraulic Valves 54
- 1.8.2. Electropneumatic Valve 82
1.9. Hydraulic Hammer 63
1.1U. Frequency Characteristics of the Assemblies 68
Cliapter 2, Dynam~c Engine Chdracteristics 73
2.1. riodel and Structural Diagrams 73
2.1.1. Structural Diagram of the Engine with Forced
Feed System 74
2.1.2. Structural Diagram of the Engine with Two-
Component Gas Generator 75
'L.2. Engine Transfer Functions 77
2.2.1. Method of Transformation of Structural Diagrams 78
2.2.2. Method of Determinants 82
2.2.3. Matr~ Method 84
2.3. Frequency Characteri~stics of the Engine $5
2.4. Stability gg
- 2.4.1. Ge� eral Stability Characteristic 88
2.4.2, rfethods of Stability Analysis 93
2.4.3. Region of Stability of the Combustion Chamber 99
2.4.4. Limits of Stability~of the Engine Systems 103
2.4.5. Stability o� ths "Engine-Regulator" System 108
Chapter 3. Wave Processes in the Liquid-Fuel Rocket Engine Lines 112
3.1. Differential Equations for Uniform Movement and Their
Integrals 112
3.2. (tarefaction Waves on Variation of the External Pressure
and L'ross Section at the Fnd of the Tube 119
3.3. Compression 4Javes on Variation of the External Pressure _
and Cross Section of the End of the Tube 12g ~
3.4. Propa~ation of Waves Through the Boundary of Pionuniform
rled ia 136
3.5. Propagation of Waves Through Discontinuities of the
Tube Cross Section 145
3.6. Propagation of Disturbances Through an Intermediate
Reservoir in the Tube 152
3,7. Propagation of Disturbances Through the Joints of
Complex Tubing 164
3.8. Propagation of Simpl.e Waves Through the Gas Line Cross -
Section Discontinuity 167
Chapter 4. Random Processes in Engines 173 �
4.1. Sensitivity of Dynamic Characteristics to the
Disturbances 173
4.2. General Characteristics of Random Processes 179
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4.3. Statistical Characteristics of Random Processes 182
4.3.1. Characteristics of Random Processes 182
4.3.2. Determination of the Statisticai Characteristics
of the Properties of the Random Process 188
4.4. Transmission of Random Signal i92
- Chapter S. Starting Up the Engine 196
5.1. General Start~Up Characteristic 196
5.1.1. Components of the Start-Up Process 196
5.1.2. Classification of Types of Start-up 19�3
5.1.3. Forcing the Start-Up Process 201
5.1.4. Requirements Imposed on the Starting Process 204
5.2. Conditions of Emergency-Free Starting 205
5.2.1. Ignition of the Components 205
5.2.2. Starting Overload of the Chamber 206 _
5.3. Power Engineering Starting Capabilities 20$ -
' S.4. Characteristic Features of Startiug Engines in a
Vacuum Undei Weightlessness Conditions 211
5.4.1. Starting in a Vacuum 211
5.4.2. Starting Under Weightlessness Conditions 212
5.5. Theoretical Calculation of the Starting Process 215
5.5.1. Peculiarities of Ca~culating Start-Up 215
5.5.2. Mathem~tical Model of the Starting Process 217
5.5.3. Calculation o~ the Starting o.f the Microengines 219
Chapter 6. Shutting Down the Engine 223
6.1. Basic Specifications of the Shutdown Process 223
6.1.1. Types of Shutdown 223
- 6.1.2. Aft~rflaning Impul~e of an Tn~;inc~ (IFD) 224
6.2, Cocn;~onents of tlie Afterflanin~ Impu].,e 226
6.3. Calculation of the Engine Aftereffect Pulse 229
6.3.1. Calculation of the Components of the IPD in
the First and Second Sections I1_2 230
6.3.2. Calculation of the Component 13 230
6.3.3. Calculation of the Component I4 234
6.4. Dispersion of the Engine Afterfiamin~ Imnul~e 240
6.5. Methods of Decreasing the Mean Value of the IPD
and Its Dispersion 241
6.5.1. Decrease in Mean Value and Dispersion of the -
IPD 242
6.5..2 Methods Excluding the Effect of the IPD on the
Stage Separation Conditions 243
Section II. Dynamic Characteristics of Solid Fuel Engines 246 -
Chapter 7. Equations of Dynamics of the Solid-Fuel Rocket
Engine Chamber 248
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7.1. Ptat~.~rial Balance Equation of the Solid-Fuel Rocket
Engine Chamber 24g
7,2, Energy Balance Equation of the Solid-Fuel Rocket
Engine Ctiamber 250
_ 7.3. Equation of Gas Inflow and Variation of the Charge
Conf lgur.st i.on 252
7.4. Gc~uattona of Heat Exchange of the Combustion Products
with the Surface of the Engine and the Charge 256 _
Ctiapeer 8. System of Equations for Determining the Combustion
Ratc of. Soli.d Rocket Fuel 262
8.1. Model of the Combustion of Solid Rocket Fuel 262
8.2. System of Differential Equations of the Combustion Rate 266
8.3. System of Linearized Equations of Solid Fuel Combustion.
Estimation of the Degree of Nonstationarity 272
8.4. Instability of Combustion in Solid-Fuel Rocket Engines 277
8.4.1. Low-Frequency Instability of Combustion in the
Solid-Fuel Rocket Engines 278
8.4.2. High-Frequency Instability of the Combustion
in the Solid-Fuel Rocket Engine 280
Chapter 9, Operation of Solid-Fue1 Rocket Engines under
Transitional Conditions 2g2
9.1. General Characteristics of the Transitional Operating
- Conditions of 9olid-Fuel Rocket Engines 282
- 9.2. Engineering Method of Calculating the Arrival of the
Solid-Fuel Rocket Engine in the Steady-State Operating
Conditions 2$4
9.3. Transient Conditions on Step Variation of the Thrust 291
9.4. ~Extinguishing of the Charge of the Solid-Fuel Rocket
Engine with a Fast Decrease in Pressure 295 ~
9.5. Extinguishing of the Charge of the Solid-Fuel Rocket '
Engine by Introducing Coolant into the Combustion
Chamber 301 -
9.6. Random Transient Conditions 305
Cliapter 10. Charge Ignition Peri.od of a Solid-Fuel Rocket Engine 307
10.1. Basic Types of Igniters and Ignition Models 307
10.2. Heat and Mass Exchange Between the Combustion Products-
of the Igniter and Charge Surface 310
i 10.2,1. Heat and Mass Exchange when Using a Jet Igniter 310
10.2.2. Heat and Mass Exchange of Using an Igniter
- with Rupturable Case . 315
_ 10.3. Frontal Ignition of Ballistite Fuel by Hot Gas , 317
10.4. Center Ignition of a Charge of Ballistite Fuel under
the Effect on Its Surface of Hot Condensed Particles 321
10.5. Frontal Ignition of Mixed Fuel by a Hot Gas Flow 326
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10.5.1. Heating of the Surface of the Mixed Fuel
During Ignition 327
10.5.2, Formation of a Reactj.ve Mixture of the Products
_ of CompQSition of the Fuel in the Gas Phase
and Its Ignition 329
10,6. Center Ignition of Mixed Fuel 331
1().7. System of Equations for Determining the Thermodynamic
Parameters of Solid Fuel Rocket Engines During *_he
Ignition Period and Joint Combustion of the Igniter and
the Charge 332
(:haptFr 11. Dynamics of the Solid-Fuel Rocket Engine Cl'~ambers as
Objects of Automated Control Systems 336
11.1. Pe^.uliarities and Possible Methods o~ Regulating the
Thrust of Solid-Fuel Rocket Engines 336
11.2. Equation of the Chamber with Adjustable Critical
Cross Section of the Nozzle 340
11.3. Equation of the Chamber with Charge Gas Form~tion
Control 348
11.4. Equations of Chambers with Thrust Adjustment by Varying
the Feed of the Additional Component 349
Bibliography 358
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PUBLICATION DATA
English title : AYNAMICS OF ROCKET MOTOR SYSTEMS
RuSSian t.itl.e , S2ATIKA I DINAMIKA RAKETNYKH
DVIGATEL'NYKH USTANOVOK V DViTICH
KNIGAKH
� :~uthor (s) ; Ye. B. Volkov, T.A Syritsyn,
G. Yu. Mazin
' Ed-ttor (s j ,
Publishing House , Mashinostroyeniye
Place of Publication , Moscow
Date of Publication , 1978
Signed to press . 15 P4ar 78
Copies , 1550
COPYRIGHT . Izdatel'stvo "Mashinostroyeniye", ~
1978
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ANNOTATI~N
= [TextJ This book is written on the problems of the theory, engineering
analy~is aud calculation of the dynamic characteristi.cs of liquid and
solid-fuel engines. Primary attention has been given to the low-frequency
dynam.ics of the engines.
A discussion is present~d of the methods of describing the dynamic
characteristics of the units and the engines as a whole and also the sensi-
tivity of the dynamic characteristics to external and internal disturbances.
- A study is made of the wave processes in the pipelines, the compression
and rarefaction wave propagation in complex lines. A static analysis of
the engine dynamics is presented on a Ferformance level. The methods of
calculating the engine characteristics under transient conditions are
given, and a detailed discussion is presented of the process of the ignition
of solid fuel in the solid-prop ellant rocket engine.
This book is designed for scientific workers and speciali~ts in the field
of rocket engine building.~It can be useful to teachers, postgraduates and
students in the advanced courses at the higher technical schools in the
corresponding specialties.
There are 32 tables, 151 f igures and 52 references.
1
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.
FOREWORD
The improvement of r.ocket engines caused by the solution of new problpms
in the f ield of rocket and space engineering is proceeding along the path
of the further complication of the schematic and structural solutions
and improvement of the parameters of the operating process.
The most complex from the point of view of the occurrence of the physical-
chemical processes are the transient and nonsteady-state operating condi-
tions of the rocket engines. At the same time the transient operating
conditions basically determine the operating reliability and the operating
stability of the engines. Ther.efore extremely great significance is
attached to the investigation of the nonsteady state processes in the theory
and practice of engine building.
At the present time there are a large number of papers in which the
~tynfim:ic:~ oC rocket engines are investigated. B. F. Glikman, Y~~. K. Moshkin,
V. A, ;taktiin, M. S. Na tanzon, and so on have made a significant contribu-
tiori to the resolution of the problems of the dynamics of liquid-fuel
rocket engines. Some of the problems of the ballistics of solid-fuel
rocket Pngines and r.onsteady state combustion of solid fuel have been
investigated in the papers by R. Ye. Sorkin, Ya. M. Shapiro, B. V. Orlov,
B. T. Yerokhin, B. A. Rayzberg, Ya. B. Zel'dovich, and so on.
However, scientific and practical experience in the field of dynaraics,
especially the dynamics of solid-fuel rocket engines, has been discussed
and generalized clearly insuff iciently. The available individual articles
and monographs are basically on special problems.
This book is one of the first in which the problems of the dynamics of
- liquid-propellant. and solid-prop~llant rocket engines and the interrelation
of the operating processes in the engine systems are discussed from unique
points of view in systematic form. r
The ltquid-fuel rocket engine is a dynamic system consisCing of a branched
network of gas and liquid lines connecting the engine power units.
The dynamic characteristics of the engine are determined to a great extent
by the characteristics of the lines; therefore a great deal of significance ~
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is attached to the wave processes in the lines in this book. On the
organizational level an effort is also made to discuss the most complex
and tl~e least lnvestigated statistical analysis of engine operating
dynamics.
The authors, understanding the complexity and the insufficient degree to
wliicli number of the problems of nonsteady state processes in engines
liave been studied, ha~�e not set the goal of exhausting this topic. The
book discusses the methods of analysis, and in some cases also synthesis,
of the low-frequency dynamics of rocket engines.
Special attention has been given to theory. The experimental data on the -
dynamic characteristics known from the literature are called on when
necessary to substantiate the mathematical models or to confirm the correct-
ness of the theoretical conclusions.
The formulas and numerical values are pres ented in accordance with the
International System of Units.
The bibliography at the end of the book is not a bibliography of the prob-
lem at large, but, with litCle exception, only a list of sources from
which information was borrowed.
_ The authors express their appreciation to Candidate of Technical Sciences
A. S. Kotelkin for the materials made available to write Chapter 3, and
they express their appreciation in advance to the readers for critical
remarks and suggestions which it is requested be sent to the following
address: Moscow, B-78, 1-y Basmannyy per., 3, izdatel'stvo Mashino-
stroyeniye.
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r r~i~~u V~/u VL~LL
SECTION I. DYNAMIC CHARACTERISTICS OF T,IQUTD-FIJEL ENGINES
~he dynamic characteristics of an engine determine the interrelation of
the working processes in the units under nonsteady-state operating condi- _
tions.
'The nonsteady-state operating conditions of the engine include the condi-
tions under which the parameters of the operating process are ti.me func-
tions, that is, _
{yf}=var, where. yi~~t~=Pw m, n ahd. so on.
The engine is characterized by several nonsteady-state operating conditions:
the starting mode, the sh u tdown mode, the transient mode (switching of
tr?e thrust stages), and regulation of the parameters of the operating pro-
cess. During the nonsteady-state modes, the structure of the engine is
~mder significant mechanical, thermal and erosion load gradienta which in
- the case of unfavorable combinations of parameters of the operating process
and bearing capacity lead to emergencies and fatlures. Therefore the
s~:udy of the behavior of the units and the engine as a whole under non-
_ steady-state conditions, that is, the ir~vestigation of the dynamic char-
acteristics, is an extremely important problem of engine building theory.
The dynamic characteristics include the following:
The dynamic equations of the relations between the garameters of the -
operating process;
The time constants, the boost factors, the frequency and transient char-
acteristics of the units and the engine as a whole; _
The characteristics of. the starting and sliutdown process; -
The quality and stability of the ad~ustment process; ~
The characteristics of the high-frequency and low-frequency vibrations. '
4 ~
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The engine in which nonsteady-state processes take place is a complex
dynamic system made up of a number of elements interconnected in a defined
way.
The dynam ic element of the engine is considered to Ue the assembly or
elemPnt having the characteristic features of the operating process which
can be described analytically.
The basic dynamic elements of the engine are the thrust chamber, the gas
generator, the turbopump~ the lines and the automation units. The
dynamic characteristics of the engine are determined by the characteris-
~ics of the elements. Therefore initially an analysis is made of the
dynamic characteristics of the individual elements and then, using special
methods, closure of the characteristics of the elements is carried out,
and the dynamic characteristics of the engine as a whole are investigated.
The initial material for analysts of the dynamic characteristics is a
system of differential equations describing the nonsteady-state operating
conditions of the units.
The set of differential equations of the units together with the compati-
_ bility conditions forms the dynamic mathematical model of the engine.
Depending on the specific problems of dynamic analysis, the type of equa~
tions of the units can be diff erent. Thus, when analyzing the starting
and shutdown processes, when the parameters:of the operating process -
vary significantly with time, nonlinear differential equations are used,
and computers are used to solve them.
For analysis of the transient processes when the parameters of the operat- -
ing process vary insignificantly, and the dynamic processes are character- ~
- ized by low frequencies, linear diff erential equations are used, that is,
the linear model of the engine. -
In subsequent chapters primary attention has been given to the low-fre-
quency engine dynamics.
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L LV1CliJ ~.~u vira.t
CHAPTER 1. DYNAMIC CHARACTERISTICS OF THE ENGINE UNITS
1.1. Thrust Chamber
1.1.1. Nonlinear Equations
Complex processes ~f preparing the fusl mix and conversion of this mix to
the products of combustion take place in the thrust chamber. When investi-
gating the dynamics of the chamber, the following processes are taken into -
account:
The accumulation of mass and variation of the internal energy of.the gas
in the half-closed volume of the chamber; .
Combustion; ;
Escape of the combustion products from the chamber cavity; �
All of the enumerated processes are interrelated in space and time, and I
their mathematical description is extremely difficult.
+
When investigating the low-frequency dynamics it is possible to make the ;
following assumptions:
For gas formed in the combustion chamber volume, the equations of state
and conservation of energy are valid;
The pressure waves are propagated in the chamber instantaneously, which
makes it possible at each point in time to take the gas pressure identical
for all points of the combustion chamber (from the in~ector head to the ~
nozzle entrance);
The convPrsion of fuel to the products of combustion is characterized by
the burn-up curve.
Since the conversion process is realized in the chambers in thousandths ~
of seconds, this process is assumed to be inertialess for significant
~ I
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simplification, that is, the fuel is converted instantaneously into the
products .of combustion.
Heat exchange of the products of combustion with the external environment
~loes not occur,
Making the indicated assumptions, the nature of variation of the pressure
in the combustion chamber pk(t) can be obtained by applying the laws of
thermodynamics for a gas in a constant volume
du = ~~pmoePdt -1N~mN~Tdt ~ .
(1) (2) (1.1)
Key: 1, formed; 2, escaping .
where du is the change in internal energy of the gas; iformed~ lescaping
are the enthalpies, respectively~ of the gas formed and the gas escaping
from the chamber; mformed~ mescaping are the masses of the gas formed in
the combustion zone and escaping through the nozzle, respectively.
When using the equation of state and the thermodynamic functions, the
internal energy is determined as follows in terms of the chamber parameters:
u=CoTKMKi j 6p, ~pTo6p+
jN~t= " R; a9= 1 R; (1..2)
~2~ x- 1 x-1
V
x= Cp ; fl1K== RT K,
v K
Key: 1, formed; 2, escaping
where Mk is the mass of the gas in the combustion~chamber volume Vk;
Tk is the average gas temperature; cP, cv are the heat capacitances of the
gas; R is the gas constant.
Af ter joint transformation of the equations (1.1) and (1.2) we obtain the
equation of the rate of pressure variation in the combustion chamber
which is the energy equation for a variable amount of gas in the chamber
volume
~~ei~l~- vK ~RT n6p ~t~ moGp ~1~ -RrHCi m�~T ~t)~� (1. 3) -
(1) (2)
Key: 1, formed; 2. escaping "
The combustion of the fuel components coming through the injectors into
the combustion chamber is characterized by a conversion law or the burn-up
curve ~y (t ) (Fig 1.1) .
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I~OK OI~ I~ I C I AI. lfS}~. t1Nl,Y t
Consiciering the burn-up functions, the arrival of the formed gasea is
deCinecl r~~ follows: ~
r~+r,
m~6p~t)=S m~(l-~')d~(z~~dt'
(1) o d~ '
Key: l, formed
where t is the process time; T' is the integration variable; tl is the
fuel ignition time; t2 is the burn-up time.
y~(L)
f
0
Z ~p t
~1
Figure 1.1. Fuel burn-up function
Key:
1. conv
The determination of the analytical function of the burn--up curve ~(t)
presents significant difficulties. Therefore on the basis of the third
assumption, the actual burn-up curve is replaced by a step function (see
Fig 1.1), assuming that on expiration of the time Tconv the fuel coming
into the chamber is instantaneously converted to the products of combustion,
that is,
~'~t)=0 for E G~t�P, ,
~ Y (f)=1 for t > t�p~l, ~
Key: 1. conv ~
Then the amount of burned fuel or gas formed Mfo~~(t) in the interval
0-t will be defined as
! (_z~p ~ ~
Mo6p _ ~ moaa ~t~ dt = ~ ~moK ~t) m~ ~t)) dt~ (1. 4 ) :
(1) o b (2) (3) i-
Key: 1. formed; 2. oxidant; 3. fuel
where mo~idant~t~~ mfuel~t) are the consumptions of the oxidizing agent i
and the combustible component of the fuel per second.
8
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The upper integration limit t-TCOnv is taken because the fuel (the oxidant
and the combustible component) entering into the chamber up to this time
still has not burned as a result of the presence of the conversion delay
Tconv�
Uifferentiating equation (1.4) with respect to the integration limit, the
gas formation rate is determined
. ' 2 /
niu P~~f~=~1 , dtPl~/TlaK~~t-Tnp~~mrlt-'Cup~~� ~1.5~
Key: See (1.4). ~ (3)
The conversion time depends on ~he quality of the mixture formation, the
propeir.ies of the fuel components, the pressure in the gas chamber and
other Factors.
The basic factor determining the conversion time is the pressure in the
combustion chamber. Accordingly, we have [20]
'C~P ~
and
dt�P Av dpK
dt p,'~}1 dt ' (1.6)
The gas flow ra~e through the nozzle is defined by the function
(t)= ~b~x)F"Pla"~t) (1.7)
Key: 1. -cr mNeT yRT~~t(t) '
If we assume that the flow rate coefficient � and the index of the adiabatic
curve X depend weakly on the pressure in the chamber, it is possible to
set the complex ~bF~r=c, which is definedin terms of the chamber param-
eters in the steady state mode:
B=~,b~x~F _ mYRT~ , - -
"p PK ' m=m~K'~~n~.
Considering the remarks that have been made, the flow rate through the
nozzle will be defined as follows:
mxcT lt) = B PK (t )
~1) YRT�~t(t) ~ ~1.8)
K;ey: l. escape
Substituting equations (1.5-1.8 and 1.6) in the initial (1.3), we obtain
the nonlinear equation of the thrust chamber
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I vK ~Tosp ~f~ pN 1~t~ fI2 ~f -'Crtp~~ `~p~~ r~ -
~ (1.9)
~ RToe~~ ) ~~n~K -'Cnp~ ~ mr ~t - ~r~~,)~ - B YRT x~Y ~E) PK ~E)) �
Uepending on the purpose of the investigation, equation (1.9) can be
conver~ed to tlie corresponding form.
Thus, for small variations of the value of Tconv~ Which occurs during
operation oF the engine under steady-state conditions (pk varies insig-
nificantly), it is possible to assume that Tconv const and dTCOnv~dt=0.
In this case, which hereafter will be considered, f:quation (1.9) assumes
the form
dpdt l,+ V B~'~RTN~t ~t) PK ~t) _
K
- VK ~T o6p ~t~ ~moK l~ -'C~P~ ll2r l~ - ~np~~. ~1..1~~
RT depends on the ratio of the fuel components K and the pressure in the
combustion chamber. However~ the last function can be,neglected, especially
at moderate pressures, and it is possible to set
RTK=RT (K~, .
where
K= 'n�" .
m~
It is possible to def ine the analytical function RT=RT(K) by the results
of the chermodynamic calculation of the combustion of specific fuel
components for various values of K and pk. -
The following functional relation can be obtained by approximation of the
calculations: ~
- RTK (K) = A1-}- ~1 sK -I- AaKz~
where A1, A2, A3 are the constant coeff icients defiued basically by the
properties of the fuel components.
Since in the combustion chamber there is a period of conversion of the
fuel components to combustion products Tconv and a f inite time t~at the
gases stay in the combustion chamber 'rstay before the escape time
~TStay=TCOnv~~ the following expressions are valid for the fitness of the
gases:
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f~bv ~f~= A2K (f - ~~~e) -I- AaK2 (t - Y3~
n6 +
RTNCT~1~=/~1`~" QtK~1-Yup~1i'A3Ka~L-Y~ ~1.11~
(2) P~4~
Key; 1, formed; 2. escape; 3. stay; 4. conv
Considering the equations (1.11) and the function K(t)=m X(t)/mful(t)
the nonlinear equation of the thrust chamber is written ~n the fo~Ilowing
Eorm:
,rrK ) x mo~ (t t,~ ) m~ (r - ~r P~ �~6
-{-V B A~ -I- Az . p-}- A3 z" ~ 1 pK =
K mr ~r - trip~ ntr ~L - T~P~ ~
= V [A~ ~12 �ioK ~f - Sn6 - t~P~ ~ A moK ~t - t~b --'OnP~
K ~r(1-'Gu6-'G~P~ 3 mr~r-'On6-'Dnp~ J X
x ~moK lt ` ~~~p~ ~ mr ~f -~np~~. ~1 .12 ~
For the steady-state operating mode pk(t)=const, m(t)=const, from the
equation (1.12) we obtain the static characteristic of the thrust chamber:
1~RT (m -}-m )
p~ m yRTK K oK r�
- Equation (1.12) is a nonlinear equation with variable coefficient; it is
used for significant variation of the parameters of the operating process
and, namely, for analysis and calculation of the atart-up and shutdown of
Lht. ~ng:lne.
1.1.2. Linearized Equation of the Chamber
For investigation of the engine dynamics in the vicinity of the steady-
state operating conditions (ad~ustment, transient processes), it is
possible to reduce equation (1.12) to the linear equation and represent it
in linearized form in relative deviations.
If we introduce the relative deviations of the variables 8y(t)=(y(t)-�y/y,
then from (1.12) we obtain
V
x K ~ ~PK \~J ~ ~PK K oPK `~K ImUK omUK1t -'Cnp~ ~
If.~Elll~ - ~up~~ + m ( aKK ~ KoK ~f - ~~P~ - 2 (
~KK l h SK X
\ 1
X ~t - Ynp -'Cn6~�
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I
r Vcc ur r ll.ltw USC. ULVLY
The last equation is divided by the coefficient for dpk, and we obtain
the linearized equation of the thrust chamber in the standard form
k.,s -dEP~ -L / �
rr 1~ dl T 8pK ~t) = K p~, ,~~SinaK 1~ -'Cnp~ ~ K pK~ m~a/13~ ~E - Y~P~ ~
~ (1.13)
~ KPK~ K ~2SK ~t - t~P~ - SI~ ~t -'rnp - ~n6~~+
Key: 1. thrust cham.ber
where 7~ vKPK is the time constant of the thrust chamber;
K.7. -
xmRTK
Kpk, x are the pressure boosting coefficients in the combustion chamber
with respect to the x-th input signal.
K K I
K m~ _ ~
P. K+,1 pK'"`~ K+~;
K dRTK
KvK, K= Z~,~K tg aK; tg ~K= dK .
.
The equation of the ratio of the fuel component
SK (t) = SmoK ~t~ - s//l~ ~E}. .1~+~
(1) (2)
Key: 1. ox=oxidant; 2. fuel [combustible component]
The ratio of the fuel component in the thrust chamber is selected from the
condition of obtaining the maximum specific thrust.
The maximum sgecific thrust is obtained for RTk=(RTk)~X; therefore the
value of BRTk/8K for K=K defined by the tangent of the slope angle of the
tangent to the curve RTk(K), is in practice equal to zero, and
i(o~, K-O.
In this case the linear equation of the thrust chamber is simplified sig-
nificantly and assumes the form
T K.a d L dt ~t a/)K ~I~ = IC pK, m~3lrioe 1~ -~n~~ -{-K p~~ mrSil2r - t~P~. (1.15 ~
1.1.3. Dynamic Characteristics of the Thrust CHamber
The equation of the thrust chamber in the Laplace transformations has the
form
(T (ij 1) ~~PK ~S) = e-st~? [K pK, rieoKam~K ~S) K pK, ~~~Smr ~S)~+ (1.16 )
Key: 1. thrust chamber
12
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where s=jw is the complex frequency.
The latter equation can be written in terms of the transfer function
~pK ls~- e SL"P ~W r~ m sl?IuK T ~~p ~ a/Tir~~ ~1..1~ ~
N oK K' C
K
where tr~ � = pK' m~
is the transfer function of the pressure in the
~'K~'"f TK.as -4- 1
ttirust chamber with respect to the i-th fuel component.
6riroK
WaK~ mo~~s~
_ fTnv sFh
e
d r'n~
WPK, m~ (f)
Figure 1.2. Structural diagram of the thrust chamber
The structural diagram of the chamber as a dynamic element constructed by
equation (1.17) is shown in Fig 1.2.
Thus, witfi respect to its dynamic structure the thrust chamber is a set of
inertial and delay elements.
The transient characteristic of the thrust chamber, that is, the variation
of the output signal dpk(t) with step variation of the input signal
dm=h�1(t); h is the magnitude of the input disturbance; obtained by
solving the equation (1.16), and it has the form
_ t-s~P1
SPK ~f)=hKoA, m1 (1-C ~ic�x J . (1.18)
i
The transient characteristic is illustrated in Fig 1.3.
The dynamic characteristics, in particular, the inert~ial properties of the
chamber, are defined by the time constant
TK~ _ ,VKPK ~ 1 ~n6� ~
Key: (1.) Thrust chamber (1~ mxRTK x (1.19
Consequently, the value of T~hrust chamber is proportional to the time
the gases stay in the chamber.
13"
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rva vrrit,tew UJC, UNLY
~Qb ~KPPK
Sir.ce irt= - , then Tthrust chamber-aRconv'
V RTK
.
wl~ere~ u-(~pb(x) ~~k7'K~-1; 1�N= ~N is the reduced length of. tlie chamber.
P
- 6vK
am~
J
h d~~ WPK r m~~s~ d PK Y
c z�p Tir ~
3Txq 1)
Figure 1.3. Transient characteristics of the thrust chamber
Key :
1. thrust chamber
For a=const the time constant is basically determined by the configuration
of the chamber. With an increase in the chamber volume Vk or a decrease
in the area of the critical cross section of the nozzle F~r the inertia
of the chamber increases (see Fig 1.4). The boost factors of the thrust
- c:hamber depend on the ratio of the fuel components. For
IC=O, KPk~ ~x 0' Kpk~~fuel-1; for K=~, KPk,nox 1' ~k~ 1�fuel-0 (Fig 1.5).
K ~m.
rK~ ~1~ ',~AK ~
K _
QK~mOK ~1~
~1b
L RS d
Q~~+' ar5 .
~PK+'"~ 2 )
LnP (2) 0 ~ 2 K
Figure 1.4. Time constant of the Figure 1.5. Boost�~factors as a _
thrust chamber as a function of funetion of the fuel component �
the reduced length and working ratio
capacity of the gas Key:
~ Key: 1. ox
1. thrust chamber 2. fuel ~
2. conv
For a conversion (delay) time constant 'rconv const, the characteristic
equation o� the chamber Tthrust chambers+l=0 has one negative root
s=-1/Tthrust chamber~ for Tthrust chamber'0; therefore the process in the
chamber is stable.
14
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However, in the general case the conversion time depends on the pressure
iii tl~e thrust chamber, which in turn leads to a flexible positive feed-
back with respect to preseure in the chamber. With an increase in pres-
sure in the chamber, the conversion time decreases, and additional ga3
formation is obtained; the so-called intrachamber instabilir_y can occur.
- Let us determine the region of stability of the combustion chamber.
Considering the function Tconv=A~Pk, the dynamic equation is obtained for
, Che thrust chamber (1.9).
After linearization of equation (1.9) the characteristic equation of the
thrust chamber assumes the form
~T~.,-Y�P�v) s-{-1=~~ (1.19a)
where ~ , = A -
�r P~
The characteristic equation (1.19, a) has one root S= -1
.
TK.~ -'G~P� V
but its sign is determined by the ratio of the terms in the denominator.
For rconv'~'Tthrust chamber the root of the equation becomes positive, and
the thrust chamber loses stability. The chamber is at the stability limit
when the root of the characteristic equation s=0.
Thus, the condition of the stability limit is
T~P 1
rK~ ~ W . (1. 20)
The satisfaction of the condition Tthrust chamber~TCOnv~~ insures
stability of the thrust chamber as a dynamic element.
1.2. Gas Generator
The operating processes of the gas generator are similar to the operating
processes of the thrust chamber. Therefore when describing the dynamics
of the gas generator it is possible to use equations obtained for the
thrust chamber.
On the basis of equations (1.9), (1.11), (1.12), the corresponding equa-
tions are written f or the gas generator:
The equation of the gas capacity of the gas generator
~I '
dt ( , - v ( RT~.o p (f) [/l:;K (t -'C;,P) m~ (t -Y,~,p) _
rr - 3~~6) , (4) (5) ~ ~1.21)
' " r.x~i ~f) mr ~t)'.
_ Key: 1. gas generator; 2. gas formation; 3. gas escape; 4. Gonv; 5. fuel;
6. ox
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r~n urrll.ttw U~C. U1VLY
_ 'flie work capacity equations of the generator gas follow:
RTr.ob~~ = I I l K~ lt -~~~y~I ~ ~ 1. Z 2~
RTr.MCT ~~~-f 4 ~f~~ ~l " Yu6'- ~uD~~'
~1~ -
Key: l, stay
The equation of the fuel component ratio. is
K' (t) = m� . (1.23)
r
When determining the gas flow rate from the gas gen~rator to the turbine ~
mT(t) it is necessary to consider the structural diagram of the engine
- and type of turbine used.
In the engines without afterburning of the generator gas, the generator gas
escape is suhcritical
f!'! - mr~1~T~ P~~ ~1.24~ _
* P V~RT '
fI' ~
In the engines with afterburning of the generator gas, jet turbines are
used, and the generaLOr gas escape takes place to the thrust chamber; in
_ this case the subcritical escape conditions are insured:
mT = mr RT~p~r - ` p ? ~�1
- _ 2 - :+1 V r
Ir 1, _ r Ptr ~ t '
1~ Ptr l= ~a ~ \P
' prr 1~ (Prr !-~p i t l x yRT~.~t r r r r ( 1. 25)
rr ~
where p1T is the generator gas pressure at the exit of the turbine guide
vane.
Consequently, the gas generator is described by the system of equations
(1.21-1.24) and (1.25).
As a result of linearization of the equations (1.21-1.24) for small devia-
tions of the parameters, we obtain the linearized equation of the gas
generator
dbp (t) ~ ~
Trr dt -E'aPrr~E)=~ Ko~, miSmi~~-YnP~~"
~,..u,r (1.26)
Kp~~~, K' ~~s~~ ~ ~E -YnP~ "SK~ lL -'Cu6 -~nP~~ ~ KD('f~ P~Tsplt 1~~�
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In operator form
~Trrs ~ ) aPrr ~s)= ~ K � -e ST"pSm! (s) T
Key: (1) gas (1) r_o ~ p~'~~
- generator , , ~1.2~~
Kvrr, K�e-st~ r(2 - e J`,~al gK~ ~S~ +.1( prr ni~sPiT ~S)�
l ~
where T~~= vrrprr is the time consr.ant of the as
x~RT g generator;
r r9r
s~- t
q~= I- 1,2-t"~+~~~p~*~prr~ i is the coefficient which depends on the
2x escape conditions .
l - ~P1:/PCC~ ;
For the critical escape conditions when
x~
~ \
lli~%Prr=P~~IPrr=(x, +~J`~-�1=const, 4,,=1.
- Ka�, z are the boost factors which are determined by the functions
K~ 1
KPrr~'"oK 4r~K~ -f' 1) ' Kprr~mr 9r~K� 1~ i
K'
Kprr~ K~ 29rR7�r tg a~.
tg afuel is determined by the ratio of the fuel components.
KPrr~K~~ 6rii(
WPzt~mi
~_ST~p BPrz
6~
wPaa ~K�
K~ K~~ .
6P~r
WPu ~ Ptt
~
Figure 1.6. Boost factor of the Figure 1.7. Structural diagram
gas generr~tor as a function of of the gas generator
the ratio of the fuel components
The ratio of the fuel components fed to the gas generator differs sharply
from stoichiometric. Therefore for a reducing gas generator
tg afuel~0 and ICPrr~ K~>0, and for the oxidizing gas generator, on the
contrary, tg afuel~0 and Kp K~0~
then TNA is stable; if TTNA0 it
will become an unstable region.
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The self-regulation coefficient a~A is determined by the moment
characteristics of the turbine and the pumps, (1 d,Np! d~tiTr
dTH~1=~ dn dn ~
The TNA is stable if aTNA'0 and, consequently, the stability condition
is the inequality
k
~1 dd1�! > dMT .
~ dn ) dn
t
Fig 1.22 shows the torques of the pumps and the turbine as a function of
the rpm. From Fig 1.22 it follows that the self-~egulation coefficient
is equal to the difference of the tangents of the slope angles of the
tangents to the curves M(n) at the investigated point .
arrt~ = tg uH tg aT�
In Fig 1.22 the point A corresponds to the stable operating conditions.
Thus, whereas at this point corresponding to the rated operating conditions,
n=n, as a result of the disturbances, the rpm n>n varies, then NLr>MP~P i
and the rpm decreases to n as a result of the formation of excess
moment of the turbine. The reverse phenomenon takes place for n0 the flow rate in the
cross section x=0 begins to decrease according to the~:law Q=Q(t). The
reservoir is connected in the vicinity of x=0.
In order to solve the equation (1.161) the initial conditions for t=0 are
P~x, ap~x' =0.
dt
For x=R
P ~x, 1)=0. .
The magnitude of the velocity at the end of the tube where the reservoir
is installed is defined by the equation
ti~~~ t~_Qo-Q (t)-QR (j)
Fo ' (1.165,a)
where Qa(t) is the liquid flow rate to the accumulator which can be
determined from the reservoir characteristics.
Let us propose that the reservoir is an elastic tank with the characteris-
t ics :
F is the area of the piston (bellows); m is tfie mass of the moving part;
ca is the rigidity of the elastic element reduced to a unit area; h is
the deviation of the elastic system from equilibrium.
The equation of motion of the piston of the reservoir is
m d~ -~-FoQh=F~ p (0, t).
The initial conditions are: t=0; h=0, h=0. The solution of the equation
(1.165)
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. va~ vr r L~,1[w UJB V1VLI
f
Ft SP~t)sinw(t-Y)dt, -
m~' o (1.166)
_ Flc
m .
Ttie liquid flow rate to the reservoir is .
Qa~t)=Fi dt . (1.167)
'I'fte so.lution oC equation (1.161) is represented in the form of traveling
cllrect and reflected waves
P~x~ t)=Pt Ct - Qol~'Ps (t-{- a~; ~1.168)
0
v~~~ t~_Pt~O~~~)-P2(~, f).
Qao
Then from the conditions (1.165, a) using the functions (1.166) and (1.167)
we have
P~~~~ r)-P2~D.t)_ Qa-Q(t) Fi r '
Qao ~Fa -F~ S ~Pi E)~'
0
pz (0, t)] cos w(1-Y) dt. (1.169)
Here, from equation (1.168) we obtain
Ps ~E) = Pi (f - al ) ~
0
9ubstituting the last function in (1.169), for pl(O,t) we have .
Pi ) -1- Pt `E - Qo ~ _ Qo - Qo (t)
~ Qao Fo
Z e (1.170)
-Fo~ `[P(E)-PiCt - a~l~ coscu(t-~t)dt.
o!
The integral in the last equation is nonzero only for T=t, for cos w(t-T)->0.
Therefore it is possible approximately to represent this integral by the
ser ies ~
` f (t) cos (l -~C) d1= f' (t) 1 _ f� ~t) 1 _
~J w2 ~4
~1.~.~1~
Considering (1.171) the equation at the boundary x=0 assumes the form
! 21 1
Pi~~)-~-Pi It- Qo Qo-Q(t) - F1 (Pi(t)-Pi (E- 2! ~1.172)
~
Qao Po Fom L \ ao
~Fo a Lpl~t)-pl (t- Qo/J
\ ~
_ b
or -
Pt ~t)= ~ PQ Ct - ao ql.
/
a_o
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~
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If we neglect the mass of the moving parts (m=0), then from (1.172) the
recurrent formulas are obtained
r
Po ~f p~l~C-n~ ~ e�~n Qo- Ct (t ) d
f,
Fo (1.173)
u
f
wtiere pv~t)=e-n~ j ear ~~q_~ ~fl -itpv-~~t)~ dt,
b~
n = FO~Q , .
FQao
In order to calculate the reservoir we assume that the liquid flow rate
varies according to a step law, with a step height at, that is,
Q(t)=Q~-atl; here tl:tcl sure and tltclosure
at (1 - e-nro) ~1 - e
P(t)=Cx~o Fo 1 _e .
-nf,
If we let tl approach zero, then from the last equation we obtain
P~t)=~o Fon -e-Rro)~1 -e-Rr~. (1.174) -
For the actual conditions ntp=(F~ca/Flpa~)t~�T, and in this case
equation (1.174) is simplified significantly, and the calculation formula
will have the form
P~:= Qo Qi 2l. (1.175)
The formula can be used for approximate calculation of the reservoir.
For the given characteristics of the main F~, p, a~, v~ and tclosure~
p~X is defined by the formula (1.164). If p~X exceeds the admissible
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value, then p~X~~ is given, and t}ie characteristics of the reservoir are
deC Inecl
_ a0 Pm ~ x. c
Ca -
21vp
and t}ie volume is defined pmax~tF~
VQ = .
ca
1,10. Frequency Characteristics of the Assemblies
As a re:~ult of tl~e interconnection of the engine and the elastic rocket,
vibrational processes occur in the er~gine assemblies which are determined
by the dynar~fc properties of the assemblies such as the natural f requency
of the vibrations, the damping and so on.
In the vibrations ~ahich occur in the rocket, the role of the engine can be
different. If pressure fluctuations occur in the combustion chamber, then
the engine serves as a source of forced vibrations of the rocket. In turn,
the roc.ket vibrations can be intensif ied by the engine as a result of the
forced vibrations of the arrival of the fuel in the chamber which lead to
fluctuations of the pressure in the chamber and the thrust. There are many
causes for the occurrence of vibrations. The basic ones of them are the
following: startup and shutdown of the engine, operation of the regulation
system, separation of the stages, longitudinal vibrations of the rocket,
and so on. -
The startup of the engine begins with sharp opening of the startup valve
which up to the time of startup separates the components in the tank aYd the
line from the en~ine cavity. With sharp opening of the valve the liquid
rusiies from the tank downward; a rarefaction ia formed in the upper layers.
The rarefaction wave is propagated with the speed of sound along the line
to the tank and, on being replaced from it, in the form of a wave of
increased pressure it returns to the flow line. The line and the liquid
have elasticity, and on variation of the pressure they form a vibrational
system.
On the occurrence of engine thrust, there is dynamic loading of the hull
of the rocket, and longitudinal vibrations of the hull and the fuel system
occur. The analogous pr~cess also occurs when the engine is shut down.
In tlie active part of the trajectory the engine is affected by different
disturbances which have a periodic nature. The frequencies of the forced
vibrations can coincide with the frequencies of the natural vibrations of
the assemblies. Here resonance phenomena occur. Thus, it is desirable to
know the frequency properties of the engine assemblies which are character-
ized by the frequency characteristics.
The frequency characteristics of the assembly are uniquely def ined by the
complex transfer function.
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!7
In order to determine the complex transfer function it is proposed that a
harmonic effect x=x1 cos wt is fed to the input of the unit, where xl is
the amplitude, and w is the angular frequency ot the effect.
At the output of the linear system in the steady-state mode, a harmonic
Eunction will also be obtained with the same frequency, but in the general
case shtfted with respect to the phase ~ and with a different amplitude
y y~ CoS (cul c~),
The amplitude yl and the phase � are determined by the properties of the
sys[em and the frequency of forcedvibrations w.
If from the harmonic signal we proceed to the complex form by substitution
of cos c~t=eJwt, then
x=xlei~e~ y=y~e~~wr+~;
The complex transfer function is the ratio of the complex form of the output
signal to the input signal
~ ~ j~~r +p) ~
lY' ~Iw~= y=.~i t _ A~w~ el~(
x xle~"
where A(a)=y~/xl is the modulus of the complex transfer function,
A(w)=mod W(~w); ~(w) is the phase shift of the output signal with respect
to the i.nput signal; �(w)=arg W(jw).
Tl~e complex transfer function is obtained from the operator function if
we replace the operator s in it by jw
W (J~)=W (S)IS-,~
The transfer function of any order can be represented by the sum of the
real and imaginary parts
IY/ (I~)= P (w) lQ
In this case the modulus (amplitude) is defined as
(w)= V P2 ('u)-}'Q~ (cu)+
and the phase Q~~~
~p(~~~)=arctg .
P
The function A(w) is called the frequency-amplitude characteristic and it
defines the capacity of the element to amplify the input signal at different _
frequencies; that is, it characterizes the pass band of the input signal.
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1'he f~lnction c~(w) is called the frequency-phase characteristic; it defines
the phase sh'_fts introduced by the element on different frequencies.
'Ihe tiodograph of the vector W(jw) on variation of the frequency from
0 to is called the phase-amplitude freq~iency characteristic. As follows
from tl~e preceding items, the engine assemblies in linear dynamics are
tl~e booster (pumps), inertial (the mains, the reservoirs, the turbine-
pump assembly), inertial with delay (the combustion chambers and the gas
~;enerntor) 1nc1 passive inte~rating elements.
Let us consider the standard frequency characteristics of the enumerated
assemblies (see Fig 1.29).
4].~IpOdrrtNUe A~4 r 2 AP x 3 fy K~(}
I NQCOC Q ROM~ q
wliu~! -~Gn a P KPni '~~w~' 0
w
~6 npz~~mppne,TNA w~oo K P A ~O
�6 W = tu
wr,ad = ; Tw ~ ~ '~cJ,o
w>>~r w 90~
~ KQnfOa [200QNUP. Q ~p �
�aacunepamop w ~ P W
wy/wl ~~t-i~.~t,y ,y,
w 90�
rH~
Q TN ~ 'P W _ 1 .
h'1,i~1: r. ( I!r W) ~r I( P ~ cu
! w) Ul.po G!- ~ �
TN~Tr
w ~
, Figure 1.29. Characteristics of the Elements
~:ey :
l. Equation
- 2. Phase-amplitude frequency characteristic
3. Phase-amplitude characteristic
4. Frequency-phase characteristic
5. Pump
h. Main, turbine-pump assembly
7. Combustion chamber, gas generator
a) Pumps ~
The pump equation
~pN 1S~-=KpH,nFll ~S~-KPn~f~m ~5~~ KPH~Pax~Psx~S~�
The complex transfer function ~/~~~,~_K
v,,,X,
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roK or�N r~;int.. us~: orr~.Y
Thus, the pump transmits the input vibrations, amplifies their amplitude
and does not change the phase.
b) ~tains, Turbine-Pump Assembly
The transfer function of the main and the turbine-pump assembly have the
form
1~/ (s)= Ky.x .
T~s + 1
'1'he complex transfer function
W ~~w~ = K1?..r _ p ~'n~ JQ ~w~+
JT y~ 1
Ky~x Ky,sT yw
~ ~ W~ I -f- Ty~us ~ Q ~w~ = - 1-I- TYu~z ;
Ky,.~
( w ) _ .
1/ ~ ry~,2 ,
~ _ - a rctg ~,T~.
In order t~ construct the frequency characteristic, A(w), ~(w), W(jw) -
are calculated for different frequencies from 0 to
In order to calculate the phase ~(w) it is convenient to use the table
- of ~=~(wt) (Table 1.2).
WT I 0 I0,05I 0,1 I 0,2I 0,5 I 1 I 2 I 5 I 10 I 20 ~ oo Table 1.2
p, rpall I 0 I 2�~0'I 5�40' 1 t�20'I26�50' 4S� Iti3�30' 78�40' 8~1�10'~87�10' 90�
1 I l I I I I I
Key;
1. degrees
As follows from analysis of the functions and the graphs (see Fig 1.29)
- the inertial element on low frequencies amplifies the input signal and
changes the phase littie. For high frequencies w>1/T vibrations occur
with sharp attenuation, that is, they are poorly transmitted. The smaller
the time constant T, the wider the frequency pass band.
- c) Combustion Chamber and Gas Generator
The transfer function of the combustion chamber and the gas generator has
the form
W( r.s+l .
S Ke s np
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. ..a a Llill~L uOG UIVLl
It is possible to represent the transfer function in the form ot the
derivative ~~f the transfer functions of the inertial and delay moments
W ~ fT`u 1 8 ~dsnp~ -
K
A(~~)= Y~ +T2W2 ; -arctgWT-~,t�N.
'Che de1~3y is an additional phase sllift by the angle wt,~p
cl) Pump in the TtJA System
The equation for the pressure at the pump included in the TNA, depending
on the flow rate of the component has the form
(TTH~s t ) aPH (S)= - i~pd,m ( t -~T�~~ am ~s).
~ The transter function
W ~~w~ ~PN~m 1 r I~HA, .
The amplitude and phase characteristics
A~~)_ ~yR,m y 1-f- THW2 �
~1~-T-
T~2 '
~1~~~= -BiCtg' ~~Te'+' TTHA~ ~
1 ~~T rT THA
'The amplitude characteristic demonstrates that the element transmits low
frequencies with amplification c�oefficient close to KpH,.. The high
frequencies are suppressed. For the middle frequencies ml/ Tg~
a negative phase shift occurs.
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CHAPTER 2. DYNAMIC EI~GINE CHARACT~RISTICS
2.1. Model and Structural Diagrams
The characteristics of an engine as a dynamic systen~ and the object of
control are uniquely defined by a system of linearii;ed equations, the
units describing the transient processes in the vicainity of the steady-
state regime. However, the system of equations does not permit establish-
menC of the qualitative relations between the individual units, which is
necessary in the preliminary design phase. _
The qualitative dynamic analysis and synthesis of an engine are cor~ven- -
iently realized using structur.al diagrams.
The structural dia~ram of the engine is the system depicting the it~ter-
relation of the dynamic elements described by the operator equations.
Ttie structural diagram depends on the pyro(pneumo) hydraulic system of the
engine, but it differs from it with respect to content, for it depicts
not the units, but the dynamic elements and the relations betwgen them.
The schematic diagram is a unique diagram for th2 given engine. The
structural diagram for the same engine can be somewhat dependent on the
method of breakdown of the engine into elements and the system of variables.
Thus, the structural diagram depicts not only the schematic diagram of the
engine, but the specif ic peculiarities of the processes in the units which
are taken into account in the mathematical model of the engine.
The initial data �or constructing the structural diagrams are the engine
diagram and its mathematical model presented in operator form. -
The symbolic notarion far the ogerations on the variables closing the -
individual elements are taken for the construction of the structural dia-
grams (see Fig 2.1).
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x2
x x x (-1
W (s)
~ x~
y= xt { x1
1) x b~ � c~
~
Figure 2.1. Symbols of the structural diagrams:
a-- element; b-- ~unction; c-- adder
2.1.1. Structural Diagram of the Engine with Pressurized Feed System
The engine equations represented in operator form are presented below.
The thrust chamber equation
apK ~S~ - W pK ~oK ~S~ e f s�FSI)1~~ ~S~ ~ W P~m e aT�PS/12~ ~5~,
K 1'
The equat ion of the lincs from the tanks to the chamber
bI11i K lS~= Wmoe�~6.oK ~s) aPe.~ K~s~ - WmuK~ PK ~S) 3Pr ~S~ -
- Wm~.e~K(S) a:. K (S)c (2.1)
bI12~ (S)= Wmr~ ~6.r ~S~ BPb.r ~S~ vKBPe ~S~ - W~~~ E~$:r ~s~�
The transf er functions are defined by the time constants and the boost
Factors
~D m! ~ KPK~'"! . W, Kmi�pK .
~ � ~~S TKS 1 ' vK TMIS 1 ~
(2.2)
K.
~m ~ a ~5~- ~r~ psr W (S = ~~,E!
1 6! 7'M!s -I- 1 '"l'E! ~ TMtS 1~
In accordance with the system (2.1) the structural diagram of the engine
has the form shown in Fig 2.2. ~
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� ~1~ . WmoK�v~~
~~O.o,~
~ Wmarr.A5.nK ~
S~cR ' ~ mox
Wm ~{ux ~ ,rr~p Bpx
e00, t
~2~ wm2.Pb:
d'}~y WPrt. mz
Wmt. E: f-1 �
Wmt . Px
Figure 2.2. Structural diagram of the engine with Pressurized feed
system
Key:
1, tank, oxidizing agent
2. tank, fuel
As follows from the structural diagram, the output signals of the engine
are the pressures in the tanks (Ptank i~ and the hydraulic drag coeffi-
cients of the lines of the thrust chamber (~i) , which can be taken as the
control inputs (the regulating parameters) . -
Positive feedback exists between the lines and the chamber, that is, the ~
pressure in the chamber depends on the flow rate of the oxidizing agent
and the cambustible fuel component, and there is negative feedback the
f low rates of the components depend on the pr.essure in the thrust chamber .
The latter relations are realized by the transfer functions of the lines
Wmi~pk, -
2.1..2. Structural Diagram of the Engine with Turo-Component Gas Generator
The mathematical model
~ ,
Lps ~5~= WpK,m~,~S~ e~~Palllo~ ~S~--~- W pK ~S~ e-~~p atlip ~S~~
� ~ ~
aPrr ~S)= Worr~~oK`S~ e ~npS/IIoK (s) Wyrr~m~ ~`ssoPamC ~S~ ~
. ~ �
�-}-Wprr~K~ ~S~ e ~nP$K,/ ~5~~ ~ ,
, . , . , (2.3) .
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SK" (s) - BmoK (s) - Sm~ (s);
~illll = ~~m PM ~ lS~ ~px t~s~ W m~ ~PK~S~ apK ls~ - W m f~ E fls~
Srit~ ~?X/m. p~S) aPNi ~S) - W,;,~ nrr (s) Sprr~s) - Wm 1,E1(S) BEr~
~ d~
bl~l~= Km~, ,~lamr (s)-~-Km~~ m'$1Ptl ~S~i
1
apH.tK ~S~ = Wpx.o' P~�~ a~I'f IS~ ~ u'~~P ~K' IS~ a~~~~ \S~ _
x.oK
- ~ ~pN.o~:~'n~K~S~ Si/1~~ ls~ ~ ~~pe.dc~mr`S~ Emr ~S~ ~ -
* ~~px.oK~ puz.ac~s~ Zpsx.oK ~5~+
,
/~N.r S~ = w Px.r~ Prr~PI'f ~SI -C~' ~px.r~^m~K~S~ E/)1uK ~S~
/ ~y (2.3)
^W px.r''nr~s~ ~mr ~S~ ~WPe.r~h~~S~ sK~ 1S~ T~Px.r~Pex.rlS~ s/~et.r~s�~
The structural diagram of the engine is presented in Fig 2.3. As follows
from the diagram, the input signals are the pressures of the input to the
pumps and the hydraulic drag coefficients of the lines to the chamber
(~i') and the gas generator (~i"). When analyzing the structural diagram,
attention is given to the presence of two groups of circuits, inside which
basically all of the feedbacks are closed; these are the "GG-TNA-CG lines"
and "thrust chamber-thrust chamber lines" circuits.
Compl.ex interaction takes place between the elements of the engine by
means of the cross positive and negative feedbacks. The flow rates of the
fuel components depend on the pressure after the pumps (the impeller rpm
of the TNA) and the pressure in the combustion chamber (the gas generator).
In turn, the magnitudes of the flow rates in ea.ch of the lines influence
the pressure created by the pump which depends on the pressure in the gas
generator. All of the f eedbacks are closed on the turbine-pump assembly
and act on the thrust chamb er only by changing the TNA impeller rpm.
Therefore, the turbine-pump assembly has defining influence on the engine
dynamics. The turbine-pum~ assembly is an inertial element with relatively
large time constanr by comparison with the other elements. Consequently,
for sufficiently large frequencies of the disturbances and the useful
signals coming from the gas generator, its rpm cannot vary, which means
that the operating conditions of the thrust chamber also cannot change.
At low oscillation frequencies of the input signals the turbiae-pump unit
can change ~pm and transmit signals to the thrust chamber. Thus, in the
engii~c the turbine-pump assecnoly is the filtering element.
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~ .
I K~_ n. n'r� ,mp.. Tp.
wmc..[~ WP~o,~~ ~ Wm~ p.
1-1
:vri~~..; J ~c,.o.o, M.v,., �'la,
woe .Or~ w/hu.. Eu.
Wpt� mc�
DOro Wn~o.m= {0e du
L.~n J R
~~0�,x~~ P`vr N,t,~^o~ va,z F' e
.Z
~?tt, mf
VlOat.Ol~t ~vmi,t'.
Wme~Dt wOr .'r~ LVm;.R~t 110~~mt
!-I
WmJ.O. '~C.:.mr~ Wrbt.On
~
1-~ xme~m Km . mi
f~~
Figure 2,3. Structural diagram of the engine with the pump
feed system
The structural diagram with afterburning of the generator gas difEers
insignificantly from the structural diagram of the engine without after-
burning. Instead of feeding the liquid component to the combustion
chamber without afterburning,.a gasified component is f ed through the gas
line with the afterburning system. This peculiarity is theoretical and
essentially has no effect on the engine dynamics. Whereas the flow rate
of the liquid component reaching the thrust chamber depends only on the
TNA impeller rpm and the pressure in the chamber, the gas parameters in
the gas line basically depend on the gas pressure and temperature in the
gas generator and depend only slightly on the TD1A impeller rpm.
Consequently, in the system with afterburning there is a direct (positive)
relation between the gas generator and the combustion chamber through the
gas line not having f iltering properties similar to the properties of the
TNA. Therefore in such engines the operating conditi.ous of the thrust
chamber are sensitive to the disturbances coming from the gas generator.
2.2. Engine Transfer Functions -
For analysis of the stability and determination of the regulating inputs
(regulating parameters) it is necessary to know the engine transfer func-
tions.
The transfer function reflects the mathematical relation between the input
and output signals and is the ratio of the Laplace transforms of the output
signal to the input
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WwX ~S)= e/ (s) .
x ( s)
Knowing Che transfer function, it is possible to write the equation for
the rela[ion
y ~S)=W
y,X ~S) x ~S)� (2. 4)
Since t:l~c~ traiisfer. Functions are ratios of polynomials
~ (S)_ bms"~ -f- bin-1�s'"-1 . . b~s -F- bo
a�s� -F~ an-~ s"~ 1-F- + a is -I- ao '
the equation of this relation can be rewritten in the form
(a~s~ a~-~S"-' . . . a,s -~-ao) +J ~S)=
_ (bm5m b~n-lS"`-'~ . . . b,s bo) x (s). ~ 2 . 5 )
m
Using the Laplace expressions y(s)= e-~y(t)dt or for the derivatives
J
Sn i an~ ~t~ from e uation 2, 5 it is ossible to
d1" ~ Q ~ ~ P proceed to the
differential equation
an d~ynt, ~"an-7 dn-l~~t~ -}-al d~~t~ a
dl dtR-1 , dt oy ~
~ ddxmt, b,n-1 d
dtr'm=I
, -I' . . . G, d t ~ bo,r
~ The last equation permits determination of the reaction of the engine,
that is, the change in the output parameter with time y(t) .for the given
� disturbance x(t).
The transfer function permits a very important characteristic of the 2ngine
to be obtained simply the boost factor
KA=1im W (s)- bo ,
j+0 a0
The analytical expression for the transfer function can be determined by
transformation of the structural diagrams, by the method of determinants
and the matrix method.
[
2.2.1. Method of Transformation of Structural Diagrams
In automatic control theory [5] a structural analysis method has been
developed which permits as complex a structural diagram as one might like
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to be converted to one equivalent element with tt~e transfer function
W ~S)=F [W~.r ~S)~�
tn c~r~l~rr tu ~irtcrmine the Cransfer function, the hranched structurnl dia-
~;r,im ir; rearranged tntc~ a slnble-loop dLagram by transfnrmations of it
according to defined rules of combining elements into one equivalent ele-
ment and carrying the elements through the adders and angles [sic].
NcxodHCA cxerr� 1 3KBuBaneNmrraA exena
_ x~~ %2 W1 YJ W3 x~ x~ W x4 W~.fl W
l~l ~
wl x,
" yyZ Xz X W y w= f wi
9
w XJ
3
X
W~ y " yy Wt
y w= i�w w
r 2
w2
" w y " w
9 y
w
x W x
9 W 9
x
~,y-1
x
X~ w X~
y y
xT uy
x2
zr ~ ~N 4 x~ w y
~ x
z w
Figure 2.4. Transformations of structural diagrams
Key:
1. Initial diagram
2. Equivalent diagram
The rules of transformation of structural diagrams are shown in Fig 2.4.
In the example of the structural dia~ram in Fig 2.2 let us demonstrate
how we obtained ~he transfer function of the pressure in the chamber with
respect to any disturbance Sptank i~ d~i�
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' - ~~+�u+ u.JL V1~L1
The transfer function is determined successively for each disturbance.
Otiiers are assumed equal to zero in this case.
'NOrc
x~s~ e- ~p aPx X~s~ e-,rt�p QPK
ft
f -1
W~ WP.C
Figure 2.5. 1'ntermediate structural Figure 2.6. Structural diagram
diagram of the engine
The structur.~l diagram in Fig 2.2 is converted to the diagram with one
adder before the delay element; as a result, we obtain the d.ia~ram in
Fig 2, 5, where
. ~
o~; S-~ moK, pKW
ph~ m~+
~~r ~S)=WM ~ P W P ~ � } ~2 . 6)
r K K' r
X(S) is the generalized disturbance;
X~S~-W ' W � aP6.oK /S W� ~p~
m~K~ Pb,oK ~xr RJOK 1~~ mOK' EOK W pK' ~OKae~K S
+W%m~~ P6.r~FK~ ~^rsPG.r~S~+Wmr~ E~WGK~ mraEr~s~~
'I'l~e structural dia~ram in Fig 2.5 j.s finally transformed into the.diagram
wttli Crcdba~k, I~ig 2,6,
where Wp.~ ~S) = WoK ~s) Wr ~s)� (2 . 7 )
W ~ is the transfer function of the open system without considering T~onv
a~id with consideration of W~~~S~=e-t~Ps WP,~(s),
The transfer function of the closed system
"'nn
e P ~2.8~
~ ~S~ ~ + m'p.~ (S) ~
The equation of the relation between any input signa.l and the output
signal has the f.orm
aP~ (S)= ~ (s) X (s), (2 . 9 ) .
where X(S) is the input signal.
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As an example, 1et us define the transfer functions for the pressure in
the combustion chamber with respect to bptank ox and d~oX and the boost
factors, ~ -
'Ct~e ini.tial transfer f.unctions will be expressed in terms of the time
cc~nst0; therefore an increase in pk leads to a decrease in Tconv~
that is, stabilization of the system.
2.4.2. Ptethods of Stability Analyais
Stability is the property of an engine to return to the initial steady-
~ state regime or a regime close to it after any departure from it as a
result of any disturbance.
Fig 2.16 shows the standard curves for the transient processes in an
unstable system (Fig 2.16, a) and a stable system (Fig 2.16, b).
If the system is unstable, then a disturbance as small as one might like in
_ it is sufficient for the diverging process of departure from the initial
state to begin.
In a stable system, the transient process caused by any disturbance damps
with time, and the system again returns to the steady-state condition.
Under actual conditions an engine ope~ates under continuously changing _
ef~ect, for example, as a result o� the effect of the speed controls and
emptying of the tanks when the .;teady-state c~nditions are in general
absent.
Considering the indicated operating conditions it is possible to present
the following, more general dafinition of engine stability. An engine,is -
stable if the output variable remains limited under the conditions of the
effect on it of disturbances of limited magnitude.
For analysis of the stability, the transfer functions of the engine are
' used.
The output parameter of the engine is related to any input parameter by the
function
- y=~ (s)�x.
(2.20)
The transfer function ~(s) is reduced to a fraction of the type
_ ~ (S)= ~ ~S~
r~!(s)' (2.21)
where M(s) is an n-degree polynomial. Substituting (2.21) in (2.20), we
obtain
1l~ (s) y=D (s) x. (2.22 )
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The solution oE this linear nonuniform equation in the general case has the
f.orm
~ (t)=y~(r)+~� (E),
(1) (2) (2.23)
Key: 1, partial; 2, transient
wliere ynartial~t) is the partial solution of the nonuniform equation with ;
~
righthand side nescribing the forced engine regime; Ytransient~t) is the
general solution of the uniform equation M(s)y=0 describing the transient
process caused by ttie disturbance x.
s~
i
9 a~ '
I
b) L~
Figure 2.15. Transient process curves:
a-- stable system; b-- unstable system
From the definition of stability it follows that a system is stable if the
transient process ytransient~t) is damping, that is, for t-~ ytransient~t)-'~� I
As is knawn, the solution to the uniform differential equation is
~
~ y~~~t)=~Cres'~, ' (2.24) f
~-i
where Ci is the integration constant determined by the initial conditions
and ttie disturbances; si are the roots of the characteristic equation
At(s)=0.
Thus, the transient process Y ra ient~t) is the sum of the components, '
the number of which is equal ~o ~ie number of roots si of the characteris- ~
tic equation M(s). In the general case the roots of the characteristic
equation si are complex and conjugate -
St,r+i=at J~t+
wtiere j=~,
_ {
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Each pair of such roots gives a component of the transient process in
equation (2.24)
~?rU ~f~=~.'~Ca~~ ~Sifl ~if ~~1~�
This time is a sine curve with amplitude that varies in time exponentially.
If al0 there are diverging oscillations.
If ai=0, then nondamping oscillations will occur. In the special case
where Si=O (the real root) the component will damp when ai0,
Thus, in the general case the transient process in the system is made up of
oscillatory and aperiodic components. The general condition of the damping
of all of the components, that is, the condition of stability of the sys-
tem is negativeness of the real parts of all the roots of the characteris-
tic equation, that is, all of the ones (and the zeros in the denominator)
of the transfer function of the closed syst~m.
If at least one root has a positive real part, it gives the diverging
component of the transient process, and the system will be unstable. The
presence of a pair of purely imaginary roots si i+l~-~ Si Will give a _
harmonic nondamping component of the transient process the system will _
be at the stability limit.
The determination of the roots of the characteristic equation, especially
for high-order systems is a complex process. Therefore in control theory
indirecr. attributes have been developed (stability criteria), which permit
determination of the stability without determining the roots of the
characteristic equation. There are algebraic and frequency stabilii.y cri-
teria.
The algebraic criteria which determine the stability conditions from the
ratios of the coefficients of the characteristic equation, for high-order
systems containing transcendental terms with factors of the type est
(characteristic for engines) are not used.
The frequency criteria of Mikhaylov and Nyquist have become widespread.
Omitting the proofs of these criteria which can be found in the literature
- on automat~~ control, let us only describe their practical use.
Mikhaylov Criterion
The stability of a system according to the Mikhaylov criterion is
determined by the behavior of the hodograph of the polynamial M(s) the
denominator of the transfer function of the closed system. _
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The imaginary variable jc~ is substituted in place of s in the polynomial
M(s) .
As a result, we obtain the complex function I
~9 U~)=f',,~ (W)-I-iQ~~ (W), ,
wliere Pr1(w) is the real part obtained from the terms of M(jw) containing ~
even powers of s; Qtq(jm) is the imaginary part obtained from the terms of
M(j~,,) with odd powers of s.
Varying the frequency w from 0 to the hodograph of M(w) in the plane
of the complex vsria hles is constructed, and the crosshatching is done
on the right side of the hodograph with encirclement of it on variation
of the frequency f rom 0 to ~(Fig 2.17). If the origin of thE coordinates
falls in the crosshatched region, the system is unstable. If M(w)=0,
that is, for any frequency it.passes through the origin of the
coordinates, the system is at the stability limit. If the hodograph M(w)
does not encompass the origin of the coordinates, then the system is
stabie.
p~ R(W)
3 / ~
Qlw) ' a,=o
P(w)
! �
Q!w) ~
/U n rlw~ C1~~ '
C Wzo
. � ~ ~ -~i i
~ ~ w- '
w~
,c/' w =o
w '
~ i
~ ~
wa0
Figure 2.17. Illustration of the Figure 2.18. Illustration of the
Mikhaylov criterion: Nyquist criterion:
1-- stable system; 2-- stability 1-- stable system; 2-- stability
limit; 3-- unstable system boundary; 3-- unstable system '
Nyquist Criterion
The Nyquist stability criterion makes it possible to determine the stability
of the closed system by the phase-amplitude frequency characteristic W(jw)
of the open system.
The hodograph W(jw)=PW(w)+jQW(w) is constructed in the plane of the complex
variables (Fig 2.18). If the phase-amplitude characteristic of the open
system jd(jw) on variation of the frequency from 0 to ~ does not encompass -
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the point (-1, 0) (this point does not fall into the crosshatched region),
then the system is sta~le. If the phase-amplitude characteristic passes
(-1, 0), then the system is at the stability limit.
The dynamic model of the engine even consider~ng the delay is correct only
Eor a natural frequency band. The stability of the engine containing any
delays for limited frequency range can be characterized by the criterion
pro,osed in rererenc~ [8].
in the complex plane a region is isolated which is bounded by the f requen-
cies w=0 and w=St (Fig 2.19) . The system is stable if all of the roots of
the characteristic equation are located in the crosshatched region.
In order to determine the stability, the hodograph NI(s) is constructed;
instead of the operator s the complex reumber s=c+jw is substituted. The
parameter s varies as follows: a) c=0, W varies from 0 to SZ, that is,
along the imaginary axis; b) for w=S2 c varies from 0 to that is,
along the straight line parallel to.the c axis.
l~ ,po~s~ �
~
.A
720
~ 540
1 360 6
180
~ ~ ~ 0~4 0,8 0 t 2 3 4 ~
Figure 2.19. Illustration of the Figure 2.20. Illustration of the -
stability criterion application of the stability cri-
terion ~
The stability conditions are as foilows:
a arg ~I~l (./~)~o~~~s= -a rg M(c j:~)Io�w'; w'=1/ T~. ~ ~
The frequency characterizing the stability limit w,~.is determined from the
expression
IWp.~ (IW)1=1.
~ ;
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. Equating equation (2.45) to one, we obtain
~ ~1/(T~~-TM)2-}-47'ZT2 K -1 Tz T2
W* TKTMy2 K K M~ N ~M)�~ (2.47)
From equation 2.47 it follows that for
I ; K - rK+T,~
YT
KT M ~ Y7'K7'nt '
On the basis of equation (2,47) and �'(w)1. For this frequency range the stability
region corresponds to Py(c~)>0 and Qy(W)0) has a
significant influence on the region of stability only for small values of
the regulator time constant.
From Fi~ 2.27 it follows that the region of stability is limited by the
boost factor of the regulator.
The admissible boost factor of the regulator installed in the gas getierator
fuel line is appreciably less than in the regulator installed in the
oxidant line. Consequently, in dynamic respects it is more efficient to
insta].1 ttie regulator pk in the oxidant line of the gas generator.
~ The regulation system changes the structure of the engine and its dynamic
characteristics. This is explained by the fact that some of the couplings
between the units are broken in order to install the regulating elements,
and new couplings introduced uy the regulator arise.
The analysis of the effect of the regulators on the dynamic characteristics
of the engine can be made by the frequency characteristics considering the
- disturbances introduced by operation of rhe regulating system. Such an
analysis is also performed and presented in reference [10].
A
A
p~5 4 ~,0 4 ,
3
7
0,:5 ~ 2 ID
1 2
o too 20o w, ~/c~l~ o ~oo
20o w,f/c ~1)
Figure 2.28. Frequency-amplitude Figure 2.29. Frequency-amplitude
characteristics of the engine characteristic of the engine with
without afterburning of the afterburning of the generator gas:
generator gas: 1-- fre uenc
q y-amplitude character-
1-- frequency-amplitude character- istic of the engine without the ~
istic of the engine without the regulator; 2-- frequency-amplitud~
regulator; 2-- frequency-amplitude characteristic of the~en~ine with
characteristic of the engine with regulator IC"; 3-- frequency-ampli-
regulator K"; 3-- frequency-ampli- tude characteristic with regulator
tude characteristic of the engine pk; 4-- frequency-amplitude
with regulator m"fuel' 4-- frequency characteristic with regulator m
-amplitude characterlstic of the Key; ~~fuel
engine with regulator pk 1. w, 1/sec
Key:
1. w,l/sec
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A
0,06
, 0,0 ; ~
I '
0,07.
2
0 100 200 u,l/r (1) ~
Figure 2.30. Fre~uency-amplitude characteristic
W "
Pkptank, ox
1-- engine without afterburning of the generator
gas; 2-- engine with afterburning of the generator ~
gas
Key:
1. w, 1/sec ~
Let us only present the basic resulCs. Fig 2.28 s~iows the frequency- -
amplitude characteristic with different regulation systems for the engine
without afterburning of the generator gas.
The analysis of the frequency-amplitude characteristic f.ndicates that the
pressure regulator has a si~nificant influence on the dynamic characteris- ~
tics of the engine.
The analysis of the frequency-amplitude characteristics shows that the
pressure regulator has a significant effect on the dynamic characteristics ,
of the engine. Resonance can appear on defined frequencies which indicates ~
that the combination of parameters is close to the stability limit.
Fig 2.29 shows the characteristics for the engine with afterburning of the
generator gas and different regulation systems.
The longitudinal stability is significantly influenced by the transfer
function W , that is the effect connected with the pressure
Pk' pinp, ox
variation at the entrance to the pump.
Fig 2.30 shows W for two types of engines. ;
Pk~ Pinp, ox ,
For the engine with afterburning of the generator gas under the correspond-
ing conditions, resonance can occur which must be considered when analyz-
ing the longitudinal stability. "
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C~iAPTER 3. WAVE PROCESSES IIZ THE LIQUID-FUEL ROCKET ENGINE LINES
3.1. Differential Equations for Uniform i4ovements and Their Integrals
The transient processes in the hydraulic lines of the ~.iquid-fuel rocket
engines are mathematically described by the partial differential equations.
For a uniform nonsteady flow of compressible ideal liquid (gas) in the
absence of external mass forces the following equations are basic:
Continuity (conservation of mass)
dt d (Q~)=~+ (3.1)
Momentum
~ ar +Q`' ax + ax (3.2)
State
. P=A_I.'BQ"+ '
(3.3)
where p, p, c are the liquid (gas) density, pressure and velocity
respectively.
Expressions (3.1), (3.2) are written in the Euler coordinate system.
In the equation of state (3.3) depicting the relation b~etween the pressure
and density, the parameters A, B and n are constants. This is valid~in
the case where the dynamic processes in the liquid (gas) are not accompanied
by significant variation of the entropy, which occurs in the pressure
range to several hundreds of bars [2]. Here the constants A and B are
expressed in terms of the speed of sound and density, respectively:
s
8= ao ecao (3.4)
nQa-I ' }`~=po- n ,
Where a~ is the speed of sound in the liquid (gas) with density and
pressure pp, PO�
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For the ideal gas A=O, and for a liquid with constant speed of sound n=1.
In particular, for water in the pressure range to 30000 bars n=7.15 [2]
and A=-3045 bars, B=3045 bars-cm3/gn, where the density p~=1 g/cm3. For
water in tt~e pressure range appreciably less than 3045 bars when the
ilensity deformation is small so that I~pl �p, from the expression (3.4)
af[er Cransformations, a linear relation is obtained between the density
and ~tie pressure p=p~+3045 np/p0, which simultaneously indicates constancy
of. tlze speed of sound with liquid deformation.
The expressions for the constants (3.4) are obtained from (3.3) in
accordance with the determination of the speed of sound so that the equation ~
of state (3.3) in implicit torm contains the relation between the pressure,
the density and the speed of sound.
The initial equations(3.1) and (3.2) are valid for the liquid flow in a
nondeformable tube of constant cross section. The consideration of the
deformation of the tube walls will be made below.
If we consider the liquid as incompressible, then from the continuity
equation we have constancy of the flow velocity along the length of the tube
(aP/at=o,ap/ax=o, 8c/ax=0), and the integral of the equa.tion of mamentum
(3.2) determines the relation between the pressure and the acceleration of
the liquid column. This integral along the tra~ectory of displacement of
tlie liquid particles is known in the literature as the Cauchy-Bernoulli
integral. In cases where the time integral of the transient process in the
line is appreciably greater than the travel time of the acoustic wave in
the investigated length of line, for analysis of the transient processes
it is possible to use this integral.
Far a compressible liquid (gas) in the general case a parameter distribution
along the tube length occurs; therefore along the tra~ectory of motion of the
particles of liquid, it is impossible to obtain the integrals of the
equations (3.1), (3.2). However, there are families of such curves in the
plane of ttie coordinates x, t along which the integrals of the equations
(3.1), (3.2) are found. For this purpose let us convert equations (3.1),,
(3.2) to the following form:
~p -~-c~ dp a2� dc =0; (3.5)
dt dz dx
~ Q.a ; ac vc l+a~ dp =0~
` dt dx ~ dx (3.6)
where under the conditions of constancy of entropy the following expressions
are valid:
dQ _ ae d p= 1 v,~ ae = ae a p= i ap
ar ap ' ar a2 ' ar ' aX ap ~ vX a2 ' aX '
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The equations (3.5) and (3.6) will be reduced by term by term addition
and subtraction to the following form respectively
~'(C~'Q,' dx +a ~ L ~t +~C+'a) dx ]=0; ~3.7)
at +~~-a)� dx -a�Q�r a~ -~-(c-a)� d~ =0. (3.8)
l dx,
In the relations obtained (3.7) and (3.8), the velocity and pressure are
expressed by the full differentials on the curves dx/dt=c+a and, conse-
quently, on these curves such integral relations are valid:
S aQ -~~=~~.s=const. (3.9)
Considering the equation of state (3.3) after integration (3.9) we f ind
2a + c -,J1.2�
n-1 (3.10)
The integrals (3.10) are called the Riemanninvariants, and the curves on _
which they are valid are called the characteristics. Znasmuch as the
form of the characteristics depends on the flow velocity and the speed of
sound, the values of which vary from point to point, through the Riema.nn
invariants the values of the liquid (gas) parameters in the general case
at the investigated point in time and at the investigated point are found
numerically. For essentially subsonic flows the characteristics can be
considered as straight lines dx/dt-l-a, where the speed of sound is
constant. For this case the density is related to the pressure linearly,
and the integrals (3.10) have the following form
PIa~Q � ~=.1~,z� (3.11)
The presented solutions have an entirely defined physical meaning: riamely,
the traveling wave in the positive direction with respect to the stationary
observer with absolute velocity c+a carries the linear combination of the
speed of sound and the flow velocity J1 from point to point without change; -
here, both the flow velocity and the speed of sound in the general case
are variable on the path of the wave. In the opposite direction the travel-
ing wave is p~opagated with abs~lute velocity c-a with respect to the same
observer and carries over the other lin~ar combination of velocities J2
without change. This means if the corresponding velocities are known on
the inside boundaries of the tube, inside the tubes the flow velocity and
the speed of sound (and other parameters) are defined in terms of the
Riemann invariants in the following way:
a=n 4 1(./i-}-?2), ~l 2~2 , ~3.12)
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� va~ va a i~,.~~ty~ U.IL' V1VL1
where
2a~ 2ay
.It n-1 -{-cl' Js=n^ ~ -~s~
al, cl are the parameters of the liquid (gas) on the left boundary of the
tube; a2, c2 are the parameters of the liquid (gas) on the right boundary
of the tube.
For essentially subsonic flows from (3.11) we have
P= Z �a�Q~~i~'?s), c= 2 (Jl-J~)~ (3.13)
where
~~=P~la�Q~-~~, .1s=Ps/a�Q-c2. ,
Inasmuch as in this case the flow velocity is appreeiably less than the
speed of sound, the characteristics of the direct and inverse directions
have the following form; respective].y:
x =z1-}-a(t-t1); ~ (3.14)
x= x~ - a (t - t~),
where xl, x2 are the coordinates of the left and right boundaries of the
tube, correspondingly; tl, t2 are the times on the corresponding boundaries
for which the liquid par_ameters and, consequently, the corresponding
Ri.emann invariants are known.
In particular, x1=0 can be taken as the coordinate of the lefthand boundary,
and x2=~,, the righthand boundary, where k is the tube length, and the
liquid flow takes place in the directi,on from the left boundary to the
right boundary.
The characteristics (3.14) under the given boundary conditions intersect
at the time t at an internal point of the tube with the coordinate x
which are determined from:
t= 2(ti-~-ts-~- X`a X' x= 2(xl'I-'xs-I-at2-afl).
~
Beginning with the above-presented physical interpretation of the values
obtained for essentially subsonic f lows, the Riemann invariants can be
represented in the following form:
x-x1
x,t p xl' t
~t- paQ ~"f'~~x,t)= aQ a --}-C~xl,t-x aXll; ,
1
P (x2 ~ t _ x2 x ~
~z = p aQ t C(x, t)= aQ a - C~aC2, t-x~ a x f.
J
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Let us set x1=0, x2=k, and in the Riemann invariants the expressions for
the corresponding parameters will be represented by expansions in the
series .limited to the linear approximation:
P(O,t- x~=P(0,1)-x apl ~ ci0,t- x~-c(O,t)-x acl ;
~ a a dt x-o ~ a a dt ,r_o
p(l, t-1--x~ - p(l, t)-1-x ap I � C(l t-~-xl = c(l, t) -
\ a a dt x-t ' a/
_1-x dc I
a d! x-t ~
Substituting these expansions in (3.13), after transformations we obtain
such expressions for the velocity and pressur.e:
~~x,~~ = pco, t)-pc1,~) +~-x dp _ x aa -
- 2ae 2pa~ at I x_r 2ea~ ar I
x_o
. ~co,t>+~~r,r> x a~ r-x a~
2 - 2a ar I X_o - 2a ar I X_r ' `
P~x~t)= P~O~t)-I-P~~~t) _ x dpI _1-x dPl .
2 ~ ar r_o a ae s_~
Qa [c(o, t)-c(t, t)~-~-Q s a~ p(j)
Z o 3E ~ 0 9f H/C x o~! M~~
1 0,14 8,6 9 0,221 8~87 21 0,274 9,26
2 ~~28 10 0,334 23 0,28 10~68
3 0,173 11,07 ll 0~242 10,925 25 0,2788 9.332
4 0,066 12 0~149 29 0,283 9,384
5 0,189 8,T7 18 0,247 9~02 33 0,286 9,934
6 0,312 1~ 0,2625 10,825 37 0,287 9,984
7 0,21 11,02 17 0,263 9,17 41 0,289 9,504
8 ~,108 19 0,273 10,73 43 0,289 10,496
Key :
1. No of the reflection
2 . c ~1) , m/sec
On the leading front of the liquid in contact with the gas in the tube,
the expresaio~for decay of the discontinuity are valid; let us denote
by pl cl the values of the pressure and the velocity in the tank before
Che h~le, p2, c2, the values of the pressure and the velocity of the
liquid in the tube. Here, at the tank-tube interface these parameters have
an even superscript, and at the liquid-gas interface (the leading edge),
_ an odd superscript (the interface of inedium 2, 3).
Thus, for the tank-tube interface on the i-th reflection of the�wave
from it running from the leading edge of the liquid along the return
characteristic, we have the system of equations:
~r+i)_a2R [c~~+1)-?~t~] ?~~~=C~~~-- P~r~
' Q2Q ' (3.44)
pir+>>_ pl _alQ~~c+>>~
Considering the equations of the flow rate and the Bernoulli equation
- from ~oint investiga~ion of (3.44) after transformation we obtain:
~i~+t) _ _ at~i -I-k2) ai~l-~'k2)2 + 2Pt+2a~l~~~ ~3.45)
w !~2 Qf~ '
where k2=sla2/s2a1, sl, s2 are the tank and tube cross sections,
respectively; i=1, 3, 5, 7,....
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, ' ~ I ~ t ? i
I~ I I~ ~ ~ ~ I
~~~II~ I ~ . . .
I II I'' ~ II'II 2,'~~I'1~~~~'~.: 'y .
� 'r ;~a
' ~ ' ' ~ ~ ~ rpaMUUa cora 2,3 (1~ -
t
J3~* )
~~t+~
J~~1 Jj~l
J ~ 2
x
Figure 3.9. Field of the characteristic of the
tube when it is filled with liquid in the tank
xey:
1. Interface of inedia 2, 3
Knowin~ the value of the velocity cl~i+l) We find the velocity of the
liquid at the entrance to the tube c2~i+lj=~1(i+l)gl~s2, the pressures
on both sides of the cross sgction discontinuity, that is, at the opening
on the side of the tank pli+l and the tube p2~i+1) according.to (3.44)
Then let us construct the invariant on the foro~ard characteristic of the
tube, along which the wave is propagated from the tank tQ.. the leading
edge of the liquid: .
f~t+l>=C~1+t) + P~t-FI) -C31+2) ~P3J+2) .
a2Q a2Q
The pressure on the interface of inedia 2, 3 is defined from the condition
- of constancy of the invariant of the return characteristic in the gas
t~+~~ ~ ct+z~
Pa = Pa-}'aaea~a .
From the bbvious co~zdition p3~i+2)_p2(i+2)~ ~3(i+2)=~2(i+2) after trans-
formations from these expressions the velocity pressure and the interface
of the media 2, 3 will be defined
� (!-F4) ~ Zi+l) ~ p3 p~~+2~_ Pa -F-a3Q3f~1t1)
C2 1~- k3 a2Q ~ 1-I- k3) ~ ~ 1-F~ k8 .
where kg=a3p3/a2p2, p3, a3 are the density and speed of sound in medium 3.
Then let us construct the invariant J2- for the reflections from the
interface 2, 3 of the wave and so on so that we have the recurrent formulas
- for the invariants �
fz!)_C31)_ P3,~ ' f(!tl)_C(1~F2)~ pZl{~1) .
2 7
a1Q ' a2Q
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_ For numerical calculations wiiere multiple reflections of the waves from ttie
cross section discontinuity takes place (the medium 1, 2), in order to
obtain the steady-state value o~ the pressure gradient at the discontinuity
(plp2-ucl2) it is impossible to use the linear approximation. With a
sufficiently high degree of accuracy it.is possible instead of (3.45) to
use the formula taking into ~ccount the nonlinearity by the third term of
the expansion of the binomial in a series
tt+i>,~, Pi -f- azQ~2l~ ~P~ -1- asQlzr~~z
C~ - a1Q(1-F' k2) ~iQ2(1 ~"k2~
so that the pressure gradient at the cross section discontinuity
, p~t+~)_ P~t+i),~,~Pi ~ a2a~~~~~2 ,
- ~'Q ~1+k~2
where the expression of the first term is the value of the velocity cli+l
in the linear approximation.
In Table 3.13 the results are presented from numerical calculations for
. the case where s1=100s2 p3=1 bar, p1=p"2=10 bars, a1=a2=103 m/sec,
p1=103 kglm3, p3=1 kg/m~, a3=300 m/sec, k3=3�10'4, k2=100, e=1, u=2�104.
The calculation was performed until the time when the invariants on the
positive characteristic coincided, which means the occurrence of the
steady-state conditions of filling the 1ine.
3.6. Propagation of Disturbances Through an Intermediate Reservoir in
the Tube
The relations obtained for calculating the parameters of the drop liquid
at the tube discontinuity on propagation of the wave will be used for the
case where the wave passes through the segment of the tube which is a
reservoir of constant cross section. In the general case the tubes
connected to this reservoir have different cross sections, and the speed
of sound of the liquid in the tubes and the reservoir is different (Fig 3.10).
Let us introduce the notation al, a2, a3 the speed of sound in the tube
to the left of the resQrvoir (region 1), in the intermediate reservoir and
in the tube to the right of the reservoir (region 3), respectively. Let
us introduce the dimensionless time t=ta2/!C, R is the le~th of the,reser-
voir, the dimensionless coordinate x=x/R. In the plane x, t the equation
of the characteristics for essentially subsonic flows have the form
dx~+dt. For convenience of calculations, the time interval Qt=2 is broken
down into n small intervals so that the elementary time step is ptn 2/n,
where n is aasumed to be even. In this ~ase in the time interval O;t;2,
r~X n+l characteristics of one family are,found, where r is the character--
istic number. Then let us ~ienote by k=1 the cross section discontinuity
at the interface of the regions 1, 2 and k=2, the discontinuity of the
cross sections at the interface of the regions 'l, 3, respectively.
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Table 3.13
d ~ c2, Pi.
P2~ 2~ s~
o~ v y/c ~/c 6ap 6ap L/c K/c
~ o
~~,1---~Z. (3) (4) (5) (6) (7)
1 0 0,9 10 1 - 0,6
2 0~0178 1.78 9,82 9,79 2~76 -
3 0,0178 2,66 9,82 1,008 - 2,56
4 0,0352 3~52 9,64 9,522 4~47 -
, 3 0,0352 4,367 9,6~ 1,01 - 4,266,
i 6 0,0522 5,22 9~478 9,208 6,14 -
25 - 18~2092 - 1,0546 - 18~1037
26 0,1865 18,65 8~135 4,fr15 19~1145 -
41 - 22~6132 - 1~0677 - 22,5064
42 0,228 22~8 7~72 2,52 23,052 -
51 - 23,9628 - 1,0717 - 23~8556
52 0,241 24~1 7~59 1~79 24,279 -
61 - 24,6416 - 1,0739 - 24,5342
62 0,247 24,7 7,53 1,41 24,846 -
71 - 25,0373 - 1,0747 - 24,9298
7'l 0,?S08 25,08 T,492 1,202 25,2 -
83 - 25,3783 - 1,0762 - 25,27
R~ 0,3~4 25,4 7,46 1,01 25,5 -
85 - 25,393 - 1,0762 - ?5,2857
86 0,25393 25,393 7,46 1,07 25.~ -
Key:
1. Reflect3on No 5. p2~ bar
2. cl, m/sec 6. J2 , m/sec
3. c2, m/sec 7. J~:, m/sec
_ 4. pl, bar
At the interface k=1 or 4=1 (first series of characteristics of the
positive direction in the reservoir) we have such initial expressions:
~Pi~'~i=2Px-E'~s~ ?i=~~~'Pi/~1i
(3.46)
, ?s= ~s - Ps~ ~1= ~zsz+ . ~
where the bar at the top means that the investigated parameter is dimen- ~
sionless: namely, J~/a2; J2_ J/a ~ J_a ~
z z a- s, P~=P~IasQ;
P2= Px/a~p; p3 = Pa~azq~ ~I = ci~as; = c~laz;
Sz = S2~Si+ = a~/a2.
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FOR OFFICIAL USE ONLI
After the transformations from (3.46) we f ind the values of the velocity
in the reservoir for the cross section k=1 for an arbitrary series of
characteristics r.
~
-cz> >+ws 1 2~1-SZ)
C~' ~ 1- s~ 2 ~ ~ + ~1 + miss) tJis~~ 2,+,r~l~:lo~ -1} , (3.47)
where the superscript in the braces denotes the number of the region to
which the value of the corresponding parameter belongs, the lower first
index indicates the number of the series of characteristics, the lower
second index denoCes the number of the tube cross section in it. In par-
ticular, the index "0" pertains to the cross sectiaiof the tube 1 at its
input on the left.
Knowing the speed in the reservoir, from (3.46) we find the speed in the
tube 1 and the corresponding pressures:
~rli= c;?iss~
P:li=~i ~~:'a-c;li~:
P%21=c%21 - ?;2~i,s,
where the invariants are known
?:2~i, z= c;~~ 1, 2- P.~~i, s~
?r,'~= c.,'b-~- P~l~/~~�
Here it is proposed that the disturbance is propagated from left to rigl,t
that is, from tube 1 to reservoir 2 and then to tube 3; therefore the
speed~a~ the pressure in Lube l~are related to each other according to the
formulas for the simple waves ~il = pl -'�o ~ cl
~ ml P~,b=p , where p+ is the
disturbed value of the pressure on the lef t end of tube 1, pp is the
undisturbed pressure in the tube.
After determination of the parameters in the circles of cross sention k=1
let us construct the invariant J~2)r 1 carried along the forward character-
istic in the reservoir from the cross section k=1 to the cross section k=2:
Ji2i = c~2i'-f - p;?i.
In the vicinities of the cross section k=2 we have the following initial
relations: - -
2Ps -f-' 2Pa Ca; ?s = ~z Ps+
,'~3 - C3 - P~~3+ C2 - C3S3+
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~ v~ ~ a~a~~u V ~Jli VL\L/.
where -
'~a =a~/as; sa= s~/s2.
: ~ -
R=~s ~3 ~
' n-6
. R=10,1s
~t =2
Rs4~t'~
X
n) K�2
P Pr~
G'~GLs'Lfi3 0
0 Q
~J!!JJlA ~
L
Figure 3.10. Propagation of the simple waves through an
intermediate reservoir in the tube. The field of the
characteristics inside the reservoir. '
After the transformations we find the speed in the tube 3 for an arbitrary
series of characteristics r:
~'8)_m3-~'~31~1 2~1-52~ {3~ 1- ~2~~ ~3 ~F8~
4 1 - S3 + ~Ut/y 53~~ ~w3 -?i-1,3T~t.1 1} ~ .
where the invariant Jr~i~ i~cnown (in particular, far the initial condi-
tions), and the invariant J~~~~ has the form:
,3
ft3~ 1.3 = Ct3~1,3 ~ pi3~ 1,3~~3�
iCnowing the~ _~need in the tube 3, we f ind the values of the other
parameters in the vicinities of the cross section k=2:
C j22 - C~3~$3~
P:~~a=J;?i-c;22; /Ji32='tF13(cr31-J;3~~,a~�
Then let us construct the invariant for the reservoir on the characteristic
of the return direction:
?rz~t= ~r22 - Pi?z�
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In the linear approximation from (3.47), (3.48) we obtain the following
expressions:
-~2) ~ i?~~,z +~i~;~o -~3~ ~3~~3j1,3 ~ ~;?i .
C~,1 = - ; C~,2 =
~ V/l'SZ LD3 S3 ~
(1) (2). (2) -(3)
Pr,l = Pi,l, pr,2=~1r,2.
The above-presented expressions are valid for the case where on the
characteristic of one family on both sides of it (bottom and top) the
Riemann invariants are different, which corresponds to the propagation of
the shock through the intermediate reservoir. For disturbance of arbitrary
shape when there are n small intervals in the time interval et=2, in
relations (3.47), (3.48) and others, it~~~ necessary to write the following
invariants in this form: instead of Jr_1 ,(3.47), (3.48) we have
J:?~~,2=C~?rt,~_Pi?~e z, and also u~il~'.o=2Pr,'o-Po~ Jr?~R.z=-Prr
Under these conditions the formulas of the linear approximations are
converted to the form:
C~=~_ 2 pr.o P0~ -~1) -~2~ - 2~p~~d- pp~
1+ m~s2 , Pr, P~, Po 1 - .
+ mtss
Correspondingly, for the cross section k=2 we have the formulas of the
linear approximation:
l � ~ f(3)
~.?i = Po 4 P�lu _Po ~ -
1 + ~~SZ , ,-~,s = - Po~w3,
-~a~_ 4 ~P�lb- Po~
Cr~ l]~" LD1S2~ 1~3 'f- SS~ ~
-~3) _ I2 - 4~3 ~Pilo - Po~
P,,s-p.,~i=Po~' ~1 +miSZ)~~3 'F'S3) ~
The above-presente expressions are valid for r;n+l, and for r